4.2: Vector Algebra
-
- Last updated
- Save as PDF
Outcomes
- Understand vector addition and scalar multiplication, algebraically.
- Introduce the notion of linear combination of vectors.
Addition and scalar multiplication are two important algebraic operations done with vectors. Notice that these operations apply to vectors in \(\mathbb{R}^{n}\), for any value of \(n\). We will explore these operations in more detail in the following sections.
Addition of Vectors in \(\mathbb{R}^n\)
Addition of vectors in \(\mathbb{R}^n\) is defined as follows.
Definition \(\PageIndex{1}\): Addition of Vectors in \(\mathbb{R}^n\)
If \(\vec{u}=\left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ],\; \vec{v}= \left [ \begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array} \right ] \in \mathbb{R}^{n}\) then \(\vec{u}+\vec{v}\in \mathbb{R}^{n}\) and is defined by
\[\begin{aligned} \vec{u}+\vec{v} &= \left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ] + \left [ \begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array} \right ]\\ & = \left [ \begin{array}{c} u_{1}+v_{1} \\ \vdots \\ u_{n}+v_{n} \end{array} \right ]\end{aligned}\]
To add vectors, we simply add corresponding components. Therefore, in order to add vectors, they must be the same size.
Addition of vectors satisfies some important properties which are outlined in the following theorem.
Theorem \(\PageIndex{1}\): Properties of Vector Addition
The following properties hold for vectors \(\vec{u},\vec{v}, \vec{w} \in \mathbb{R}^{n}\).
- The Commutative Law of Addition \[\vec{u}+\vec{v}=\vec{v}+\vec{u}\nonumber \]
- The Associative Law of Addition \[\left( \vec{u}+\vec{v}\right) +\vec{w}=\vec{u}+\left( \vec{v}+\vec{w}\right)\nonumber \]
- The Existence of an Additive Identity \[\vec{u}+\vec{0}=\vec{u} \label{vectoridentity}\]
- The Existence of an Additive Inverse \[\vec{u}+\left( -\vec{u}\right) =\vec{0}\nonumber \]
The additive identity shown in Equation \(\eqref{vectoridentity}\) is also called the zero vector , the \(n \times 1\) vector in which all components are equal to \(0\). Further, \(-\vec{u}\) is simply the vector with all components having same value as those of \(\vec{u}\) but opposite sign; this is just \((-1)\vec{u}\). This will be made more explicit in the next section when we explore scalar multiplication of vectors. Note that subtraction is defined as \(\vec{u}-\vec{v} = \vec{u}+\left( -\vec{v} \right)\).
Scalar Multiplication of Vectors in \(\mathbb{R}^n\)
Scalar multiplication of vectors in \(\mathbb{R}^n\) is defined as follows.
Definition \(\PageIndex{2}\): Scalar Multiplication of Vectors in \(\mathbb{R}^n\)
If \(\vec{u}\in \mathbb{R}^{n}\) and \(k\in \mathbb{R}\) is a scalar, then \(k\vec{u}\in \mathbb{R}^{n}\) is defined by \[k\vec{u}=k\left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ] = \left [ \begin{array}{c} ku_{1} \\ \vdots \\ ku_{n} \end{array} \right ]\nonumber \]
Just as with addition, scalar multiplication of vectors satisfies several important properties. These are outlined in the following theorem.
Theorem \(\PageIndex{2}\): Properties of Scalar Multiplication
The following properties hold for vectors \(\vec{u},\vec{v}\in \mathbb{R}^{n}\) and \(k,p\) scalars.
- The Distributive Law over Vector Addition \[k \left( \vec{u}+\vec{v}\right) = k\vec{u}+ k\vec{v}\nonumber\]
- The Distributive Law over Scalar Addition \[\left( k + p \right)\vec{u} = k \vec{u}+p \vec{u}\nonumber\]
- The Associative Law for Scalar Multiplication \[k \left( p \vec{u}\right) = \left(k p \right)\vec{u}\nonumber\]
- Rule for Multiplication by \(1\) \[1\vec{u}=\vec{u}\nonumber\]
- Proof
-
We will show the proof of: \[k \left( \vec{u}+\vec{v}\right) = k \vec{u}+ k \vec{v}\nonumber\] Note that: \[\begin{array}{ll} k \left( \vec{u}+\vec{v}\right) & =k \left [ u_{1}+v_{1} \cdots u_{n}+v_{n}\right ]^T \\ & = \left [ k \left( u_{1}+v_{1}\right) \cdots k \left( u_{n}+v_{n}\right) \right ]^T \\ & = \left [ k u_{1}+ k v_{1} \cdots k u_{n}+ k v_{n}\right ]^T \\ & = \left [ k u_{1} \cdots k u_{n} \right ]^T + \left [ k v_{1} \cdots k v_{n} \right ]^T \\ & = k \vec{u}+k \vec{v} \\ \end{array}\nonumber\]
We now present a useful notion you may have seen earlier combining vector addition and scalar multiplication
For example, \[3 \left [ \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right ] + 2 \left [ \begin{array}{r} -3 \\ 0\\ 1 \end{array} \right ] = \left [ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right ].\nonumber \] Thus we can say that \[\vec{v}= \left [ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right ]\nonumber \] is a linear combination of the vectors \[\vec{u}_1 = \left [ \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right ] \mbox{ and } \vec{u}_2 = \left [ \begin{array}{r} -3 \\ 0\\ 1 \end{array} \right ]\nonumber \]